Stasheff polyhedra as moduli spaces

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\underline{Stasheff polyhedra as moduli spaces}
 
This talk is about Stasheff polyhedra. The plan would be to avoid
generalities (operadic,
Boardman-Vogt resolution) and instead focus on the specific realizations
of these
polyhedra which are relevant for Fukaya categories. The only prerequisite
is some
basic knowledge of Deligne-Mumford spaces. A useful reference is
[Fukaya-Oh, Zero-loop open string in the cotangent bundle and Morse
homotopy].
Other aspects of Stasheff polyhedra are covered in the topology
literature, such as
[Stasheff, $H$-spaces from a homotopy point of view] or
[Markl-Shnider-Stasheff, Operads in algebra, topology and physics].
 
\underline{Plan}
 
{\it Definition.} We could start by looking at the combinatorial structure
of the
Stasheff polyhedra. Vertices are given by all possible complete
bracketings, and
partial bracketings describe the cells. Low-dimensional examples can be
described
explicitly as polyhedra. The operadic structure should be mentioned.
Algebras over
the operad of chains on the Stasheff polyhedra are equivalent to
$A_\infty$-algebras
(it's probably not worth while going into details about this fact).
 
{\it Metrized trees.} The first interpretation, mainly useful for Morse
theory
(see for instance [Fukaya-Oh, op. cit.] or [Cohen-Norbury, Morse field
theory])
is as moduli spaces of metrized ribbon trees. This explains the inductive
structure of the boundary (each boundary face is a product of two
associahedra)
in terms of degenerations (trees with an infinitely long edge). 
 
{\it Pointed discs.} We first want to see the point-set description of
Stasheff
associahedra as pointed discs (with their degenerations to stable discs).
An
important consequence is that one immediately sees that these spaces are
contractible (since the interiors are configuration spaces of points on
the
line, essentially). More importantly, they are naturally smooth manifolds
with
corners. To see that, one embeds them into the real parts of
Deligne-Mumford
spaces $\bar{\mathcal{M}}_{0,n}$, so we need a review of those spaces as
well.
The relation is explained in [Fukaya-Oh, op. cit.] with added remarks in
[Seidel, Fukaya categories and Picard-Lefschetz theory, Ch. 2].
 
\underline{Notes} 
 
There are many interesting relatives of Stasheff associahedra. The closest
relative
is the Fulton-Macpherson compactification of the configuration space of
points
on the line, which actually contains Stasheff polyhedra as its boundary
face.
Another one are multiplihedra, recently geometrically realized by
[Mau-Woodward,
Geometric realizations of the multiplihedron and its complexification].
 
\underline{Dependencies}
 
Only depends on knowing basic facts about $A_\infty$-structures.
 
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